The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face

Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system. In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... Jump to: navigation, search A triakis tetrahedron is a catalan solid which looks a bit like an overinflated tetrahedron. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In crystallography, the cubic crystal system (or isometric crystal system) is the most symmetric of the 7 crystal systems. ...

Chiral tetrahedral symmetry

T, 332, or 23, of order 12 - chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D₂ or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A₄, the alternating group on 4 elements. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics an alternating group is the group of even permutations of a finite set. ...

The conjugacy classes of T are: Jump to: navigation, search In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

• identity
• 4 × rotation by 120° clockwise (seen from a vertex)
• 4 × rotation by 120° anti-clockwise (ditto)
• 3 × rotation by 180°

The rotations by 180°, together with the identity, form a normal subgroup of type Dih₂, with quotient group of type Z₃. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

Achiral tetrahedral symmetry

T_d, *332, or of order 24 - achiral or full tetrahedral symmetry. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S₄ () axes. T_d and O are isomorphic as abstract groups: they both correspond to S₄, the symmetric group on 4 objects. T_d is the union of T and the set obtained by combining each element of O T with inversion. See also the isometries of the regular tetrahedron. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... Jump to: navigation, search For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

Apart from text and colors, this volleyball has approximately symmetry T_h. The left and right side of the face with text correspond to two of the rectangles and pentagons in the description.

The conjugacy classes of T_d are: Image File history File links Photograph of a beach volleyball in sand, by photographer Douglas Bishop of itsawebguy. ... Image File history File links Photograph of a beach volleyball in sand, by photographer Douglas Bishop of itsawebguy. ...

• identity
• 8 × rotation by 120°
• 3 × rotation by 180°
• 6 × reflection in a plane through two rotation axes
• 6 × rotoreflection by 90°

Pyritohedral symmetry

T_h, 3*2, or , of order 24 - pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S₆ () axes, and there is inversion symmetry. T_h is isomorphic to T × Z₂: every element of T_h is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D_2h (that of a cuboid), of type Dih₂ × Z₂ = Z₂ × Z₂ × Z₂ . It is the direct product of the normal subgroup of T (see above) with C_i. The quotient group is the same as above: of type Z₃. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. In anatomy, the cuboid bone is a bone in the foot. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron [1], which is similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes. Jump to: navigation, search The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on...

The conjugacy classes of T_h include those of T, with the two classes of 4 combined, and each with inversion:

• identity
• 8 × rotation by 120°
• 3 × rotation by 180°
• inversion
• 8 × rotoreflection by 60°
• 3 × reflection in a plane

The full tetrahedral group T_d with fundamental domain

The pyritohedral group T_h with fundamental domain

Solids with full tetrahedral symmetry

Platonic solid: In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting only on each of its vertices. ...

Name	Picture	Faces	Edges	Vertices	Edges per face	Faces meeting at each vertex
tetrahedron	( Animation) Jump to: navigation, search For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Download high resolution version (643x607, 26 KB)Tetrahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Tetrahedron. ...	4	6	4	3	3

Archimedean solid: In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...

(semi-regular: vertex-uniform)

Name	picture	Faces		Edges	Vertices	Vertex configuration
truncated tetrahedron	( Video)	8	4 triangles 4 hexagons	18	12	3,6,6

Catalan solid: The truncated tetrahedron is an Archimedean solid. ... Download high resolution version (867x773, 50 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made using POV-Ray. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... A rhombic dodecahedron In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. ...

(semi-regular dual: face-uniform)

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face polygon
triakis tetrahedron	( Video)	truncated tetrahedron	12	18	8	isosceles triangle

Tetrahemihexahedron